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This issue could be caused by a number of reasons, including incorrect theme coding or a third-party app conflicting with the checkout page, it'd be best if you can share us your store URL. Meanwhile, you can try these steps to troubleshoot your issue.
Disable all third-party apps and see if the drop-down appears. If it does, re-enable the apps one by one to identify the source of the conflict.
Check the checkout.liquid file in the theme editor for any code that could be blocking the drop-down from appearing.
Try switching to a different theme temporarily to see if the issue persists.
Hope this is helpful for you, don't be hesitate to reach out to us if you need any further support. And it'll encourage us to dig in and support more if you can hit the Like button or mark this as Accept Solution, it means a lot to us.
Looking forward for your reply.
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